Quantum Bound States on Some Hyperbolic Surfaces

Abstract

The aim of this work is to highlight results of energy eigenstates on some noncompact finite hyperbolic surfaces. Such systems are known to exhibit both continuous and discrete spectra and are dependent on the subgroups of the modular group that underlie these surfaces. We study explicitly the cases of Maass cusp forms on the singly punctured two-torus and the triply punctured two-sphere for their eigenvalues. The eigenvalues for the torus system are doubly degenerate while for the sphere case, the eigenvalues are nondegenerate. We also note that the lowest eigenvalue of the sphere system is larger than that of the torus system.